Simplify Algebraic Expressions: Step-by-Step

Hey guys, let's dive into the awesome world of simplifying algebraic expressions! Today, we're tackling a cool problem that will make you a math whiz in no time. We're given the expression $\frac{1}{3}(9 x-15)+4\left(x+\frac{1}{2}\right)$, and we're going to walk through the simplification steps together. Our goal is to find the final result after combining like terms. So, buckle up, grab your pencils, and let's get this math party started!

Understanding the Expression and Initial Steps

First off, let's look at the expression we're working with: $\frac{1}{3}(9 x-15)+4\left(x+\frac{1}{2}\right)$. It might look a little intimidating with all those fractions and parentheses, but trust me, it's totally manageable. The first step in simplifying this expression is to distribute the numbers outside the parentheses to the terms inside. This is where the distributive property comes into play, which is a fundamental concept in algebra. Remember, the distributive property states that $a(b+c) = ab + ac$. We'll apply this property twice in our expression.

So, for the first part, $\frac{1}{3}(9 x-15)$, we multiply $\frac{1}{3}$ by $9x$ and then by $-15$. This gives us $\frac{1}{3}(9x) - \frac{1}{3}(15)$. Performing the multiplication, we get $3x - 5$. This is our first simplified chunk.

Now, for the second part, $4\left(x+\frac{1}{2}\right)$, we distribute the $4$ to both $x$ and $\frac{1}{2}$. So, we have $4(x) + 4\left(\frac{1}{2}\right)$. Multiplying these out, we get $4x + 2$.

Putting it all together, our expression now looks like this: $3x - 5 + 4x + 2$. See? We've already made significant progress! The initial distribution step is crucial because it helps us break down a complex expression into simpler terms that are easier to work with. It's all about tackling one piece at a time, and before you know it, you'll have it all figured out. This initial distribution is the foundation for the next step, which involves combining those pesky like terms. So, pat yourself on the back – you've successfully navigated the first hurdle!

Combining Like Terms: The Magic Step

Alright, mathletes, we've successfully distributed and transformed our initial expression into $3x - 5 + 4x + 2$. Now comes the really fun part: combining like terms! This is where we gather all the terms that have the same variable raised to the same power, and also combine the constant terms (the numbers without any variables). Think of it like sorting your M&Ms – you group all the red ones together, all the blue ones together, and so on. It makes everything so much neater and easier to understand.

In our expression, $3x - 5 + 4x + 2$, we have two types of terms: terms with 'x' and constant terms. Let's identify them. The terms with 'x' are $3x$ and $4x$. These are our 'x' buddies! We can add them together: $3x + 4x = 7x$. Easy peasy, right? This is the core of combining like terms – you just add or subtract their coefficients (the numbers in front of the variables).

Next, let's look at our constant terms. These are the numbers without any 'x' attached to them. In our expression, they are $-5$ and $+2$. We combine these by performing the arithmetic operation: $-5 + 2 = -3$. Remember to pay close attention to the signs! It's super important not to mix up addition and subtraction, or positive and negative numbers.

So, by combining the 'x' terms, we got $7x$, and by combining the constant terms, we got $-3$. Now, we put these two results back together to form our final, simplified expression. This gives us $7x - 3$. Voila! We've successfully combined all the like terms and arrived at our final answer. This process is incredibly powerful because it reduces complex expressions to their simplest form, making them much easier to analyze and work with in further mathematical contexts. It's the kind of skill that will serve you well, no matter where your math journey takes you. Keep practicing, and you'll be a master of combining like terms in no time!

The Final Result and Why It Matters

So, after all our hard work distributing and combining like terms, we've arrived at the simplified expression $7x - 3$. This is the result of simplifying the original expression $\frac{1}{3}(9 x-15)+4\left(x+\frac{1}{2}\right)$. It's amazing how a complex-looking problem can be boiled down to such a neat and tidy answer, right? This final result, $7x - 3$, is mathematically equivalent to the original expression, meaning that if you were to plug in any value for 'x' into both expressions, you'd get the exact same answer. That's the beauty of algebraic simplification – it preserves the value of the expression while making it much easier to understand and use.

Why does this matter, you ask? Well, simplifying expressions is a fundamental skill in algebra that pops up everywhere. Whether you're solving equations, working with functions, or tackling more advanced mathematical concepts, you'll constantly be simplifying expressions. It's like learning the alphabet before you can read a book. This skill allows us to see the core structure of mathematical relationships more clearly. For instance, if we needed to find the value of the expression for a specific 'x', using $7x - 3$ would be way faster than plugging it into the original, more complicated form.

Furthermore, understanding simplification helps build a strong foundation for problem-solving. When you can efficiently simplify expressions, you're better equipped to tackle word problems and real-world applications of math. You can translate a situation into an algebraic expression, simplify it to its essence, and then solve for the unknown. It's a powerful chain of skills that opens up a lot of doors. So, the next time you see a tangled algebraic expression, don't sweat it! Remember the steps: distribute, distribute, distribute, and then combine those like terms. You've got this, and the result, like our $7x - 3$, will be beautifully simple and incredibly useful. Keep practicing, keep exploring, and you'll become an algebra ace in no time! Happy calculating, everyone!